I'm trying to solve the following linear PDE, that is fourth order in space and second order in time:
$$ \text{(PDE):} \hspace{5mm} \frac{\partial^4 y}{\partial x^4} + \frac{\partial^2 y}{\partial t^2} - \lambda e^{i\omega t}\frac{\partial^2 y}{\partial x^2} = 0 $$
Subject to initial and boundary conditions:
$$ \hspace{-8.5mm}\text{(IC):} \hspace{5mm} 0 = y(x, 0) = \frac{\partial y}{\partial t}(x, 0) \\ \hspace{-11mm}\text{(BC1):} \hspace{5mm} 0 = y(0, t) = \frac{\partial y}{\partial x}(0, t) \\ \hspace{-4mm}\text{(BC2):} \hspace{5mm} 0= \frac{\partial^2 y}{\partial x^2}(1, t) = \frac{\partial^3 y}{\partial x^3}(1, t) $$
Here $\lambda$ is a positive parameter and the domain for this system is: $x\in[0, 1]$, $t\in[0, \infty)$.
I have attempted many different approaches for solving this system analytically but haven't been able to make much progress. I believe that no seperable solution exists. I have tried playing around with integral transform methods, but for example a Laplace transform leads to a delay-like differential equation that seems harder to solve: $$ 0 = \frac{d^4Y}{dx^4}(x, s) + s^2Y(x, s) - \lambda\frac{d^2Y}{dx^2}(x, s-i\omega) $$
I wanted to know if anyone had experience in solving a system like this and what approaches (if any) may actually yield analytical progress. Any insight/help would be very much appreciated. In particular, I am not aware if this problem would fall under the scope of some complex analytic methods.
For context, this is a dimensionless system modelling a clamped-free, elastic-magnetic beam deforming under an oscillatory magnetic field. Since the equation is linear, the presence of the $e^{i\omega t}$ term was included in the usual way where the real part is being assumed and would be taken after solving the system.